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Home , Fibonacci, Liber Abaci

FIBONACCI ARE FASCINATING, AND THAT’S NO FIB! George Soliman Instructor Raritan Valley Community College Two Truths and a Fib

1. The Sequence was discovered by Leonardo Fibonacci. FIB! th 2. The 6 Fibonacci , F6, is the infinity symbol rotated 90°. TRUTH! ( = 8) 3. There exist explicit formulas for . 𝐹𝐹6 TRUTH! (stay tuned!) 𝐹𝐹𝑛𝑛 It All Began in Ancient

• The Fibonacci Sequence seemingly first appeared in the (the art of prosody) Chandaḥśāstra • Written by the ancient Indian mathematician and Sanskrit grammarian Pingala sometime between 450-200 BC

• Describes the rhythm, stress, and intonation of speech, an important ancient Indian ritual Leonardo (6 Centuries Later) Fibonacci

• c. 1170 – c. 1250

• Known as Leonardo Pisano or simply Leonardo of , since he was from Pisa

• Given the name “Fibonacci”, which is a contraction of “Filius Bonacci” (Latin for “son of Bonacci”) in 1838 by the Franco-Italian historian Guillaume Libri

• The Fibonacci Sequence was given its name in May 1876 by the French number theorist François Édouard Anatole Lucas Hindu-Arabic > Roman

• Around 1190, Leonardo traveled to Bugia, (current-day ) with his father, where he learned the Hindu-Arabic numeral system and computation methods (0-9, place value)

• Up until then, Europeans were using for computations (tally system for recording numbers)

• He traveled extensively throughout northern Africa and the Middle East around the Mediterranean coast to study the various systems then being used

• Around 1200, he returned home to Pisa

• He realized the many advantages and conveniences of the Hindu-Arabic numeral system over Roman Numerals then being used in

• In 1202, Leonardo published his pioneering book entitled Liber Abaci (Latin for “The Book of Calculation”), which introduced the Hindu-Arabic numeral system and arithmetic methods to Europe

• The book was so influential that by the end of the 16th century, most of Europe had adjusted to the current Hindu-Arabic system

• He eventually wrote three more books Immortal Rabbits: Problem

• One of the many arithmetic problems in Leonardo’s Liber Abaci is the “Problem of the Rabbits”: • “A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also.” (Sigler 404) Immortal Rabbits: Solution • Each newborn pair of (immortal) beginning 1 rabbits, a male and a female, matures in first 2 one month and then starts to breed second 3 • “You can indeed see in the margin how third 5 we operated, namely that we added the first number to the second, namely the fourth 8 1 to the 2, and the second to the third, fifth 13 and the third to the fourth, and the sixth 21 fourth to the fifth, and thus one after another until we added the tenth to the seventh 34 eleventh, namely the 144 to the 233, eighth 55 and we had the abovewritten sum of rabbits, namely 377, and thus you can ninth 89 in order find it for an unending number tenth 144 of months.” (Sigler 404-405) eleventh 233 end 377 Immortal Rabbits: Closer Look

• Let r represent a pair of Month Population Adults Newborns Total newborn rabbits Beginning R 1 0 1 1 Rr 1 1 2 2 RrR 2 1 3 • Let R represent a pair of 3 RrRRr 3 2 5 adult (“mature”) rabbits 4 RrRRrRrR 5 3 8 5 RrRRrRrRRrRRr 8 5 13

⋮ ⋮ ⋮ ⋮ ⋮ Recursive Definition and the First 30

• = 1 1 1 11 89 21 10946 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 𝐹𝐹1 2 1 12 144 22 17711 • = 1 3 2 13 233 23 28657 4 3 14 377 24 46368 𝐹𝐹2 5 5 15 610 25 75025 • = + , 3 6 8 16 987 26 121393 7 13 17 1597 27 196418 𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛−1 𝐹𝐹𝑛𝑛−2 𝑛𝑛 ≥ 8 21 18 2584 28 317811 • Some define = 0 9 34 19 4181 29 514229 10 55 20 6765 30 832040 𝐹𝐹0 Fibonacci Spiral SOME COOL PROPERTIES! The good stuff! Sum of First n Fibonacci Numbers

= : 1 = 1 1 1 11 89 21 10946 𝒏𝒏 𝟏𝟏 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 = : 1 + 1 = 2 2 1 12 144 22 17711 𝒏𝒏 𝟐𝟐 3 2 13 233 23 28657 = : 1 + 1 + 2 = 4 4 3 14 377 24 46368 𝒏𝒏 𝟑𝟑 5 5 15 610 25 75025 = : 1 + 1 + 2 + 3 = 7 6 8 16 987 26 121393 𝒏𝒏 𝟒𝟒 = : 1 + 1 + 2 + 3 + 5 = 12 7 13 17 1597 27 196418 8 21 18 2584 28 317811 𝒏𝒏 𝟓𝟓 9 34 19 4181 29 514229 General Formula: 𝒏𝒏 = ⋮ 10 55 20 6765 30 832040 � 𝑭𝑭𝒊𝒊 𝑭𝑭𝒏𝒏+𝟐𝟐 − 𝟏𝟏 𝒊𝒊=𝟏𝟏 Proof (Mathematical Induction)

General Formula: 𝑛𝑛 = 1 𝑘𝑘+1 = 𝑘𝑘 +

� 𝐹𝐹𝑖𝑖 𝐹𝐹𝑛𝑛+2 − � 𝐹𝐹𝑖𝑖 � 𝐹𝐹𝑖𝑖 𝐹𝐹𝑘𝑘+1 𝑖𝑖=1 𝑖𝑖=1 𝑖𝑖=1 = : = 1 = 1 = 2 1 = 1 = 1 +

𝒏𝒏 𝟏𝟏 𝐹𝐹1 𝐹𝐹3 − − 𝐹𝐹𝑘𝑘+2 − 𝐹𝐹𝑘𝑘+1 = Suppose that 𝑘𝑘 = 1 𝑭𝑭𝒌𝒌+𝟑𝟑 − 𝟏𝟏 𝑖𝑖 𝑘𝑘+2 � 𝐹𝐹 𝐹𝐹 − QED 𝑖𝑖=1 for some integer 1.

𝑘𝑘 ≥ Sum of First n Even Indices

= : 1 = 1 1 1 11 89 21 10946 𝒏𝒏 𝟏𝟏 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 = : 1 + 3 = 4 2 1 12 144 22 17711 𝒏𝒏 𝟐𝟐 3 2 13 233 23 28657 = : 1 + 3 + 8 = 12 4 3 14 377 24 46368 𝒏𝒏 𝟑𝟑 5 5 15 610 25 75025 = : 1 + 3 + 8 + 21 = 33 6 8 16 987 26 121393 𝒏𝒏 𝟒𝟒 = : 1 + 3 + 8 + 21 + 55 = 88 7 13 17 1597 27 196418 8 21 18 2584 28 317811 𝒏𝒏 𝟓𝟓 9 34 19 4181 29 514229 General Formula: 𝒏𝒏 = ⋮ 10 55 20 6765 30 832040 � 𝑭𝑭𝟐𝟐𝟐𝟐 𝑭𝑭𝟐𝟐𝟐𝟐+𝟏𝟏 − 𝟏𝟏 𝒊𝒊=𝟏𝟏 Sum of First n Odd Indices

= : 1 = 1 1 1 11 89 21 10946 𝒏𝒏 𝟏𝟏 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 = : 1 + 2 = 3 2 1 12 144 22 17711 𝒏𝒏 𝟐𝟐 3 2 13 233 23 28657 = : 1 + 2 + 5 = 8 4 3 14 377 24 46368 𝒏𝒏 𝟑𝟑 5 5 15 610 25 75025 = : 1 + 2 + 5 + 13 = 21 6 8 16 987 26 121393 𝒏𝒏 𝟒𝟒 = : 1 + 2 + 5 + 13 + 34 = 55 7 13 17 1597 27 196418 8 21 18 2584 28 317811 𝒏𝒏 𝟓𝟓 9 34 19 4181 29 514229 General Formula: 𝒏𝒏 = ⋮ 10 55 20 6765 30 832040 � 𝑭𝑭𝟐𝟐𝟐𝟐−𝟏𝟏 𝑭𝑭𝟐𝟐𝟐𝟐 𝒊𝒊=𝟏𝟏 Sum of Squares of Consecutive Fibonacci Numbers + = : + = 1 + 1 = 2 2 2 1 1 2 12 𝟐𝟐 1 𝒏𝒏 𝟏𝟏 𝐹𝐹1 𝐹𝐹2 𝒏𝒏 𝑭𝑭𝒏𝒏 𝑛𝑛 𝒏𝒏𝑛𝑛+𝑭𝑭𝒏𝒏1 = : + = 1 + 4 = 5 2 1 𝐹𝐹 𝐹𝐹2 1 2 2 𝒏𝒏 𝟐𝟐 𝐹𝐹2 𝐹𝐹3 3 2 3 4 = : + = 4 + 9 = 13 4 3 4 9 2 2 𝒏𝒏 𝟑𝟑 𝐹𝐹3 𝐹𝐹4 5 5 5 25 = : + = 9 + 25 = 34 2 2 6 8 6 64 𝒏𝒏 𝟒𝟒 𝐹𝐹4 𝐹𝐹5 = : + = 25 + 64 = 89 7 13 7 169 2 2 8 21 8 441 𝒏𝒏 𝟓𝟓 𝐹𝐹5 𝐹𝐹6 General Formula: + = 9 34 9 1156 ⋮ 𝟐𝟐 𝟐𝟐 + 10 55 10 3025 𝑭𝑭𝒏𝒏 𝑭𝑭𝒏𝒏+𝟏𝟏 𝑭𝑭𝟐𝟐𝟐𝟐+𝟏𝟏 ∀𝒏𝒏 ∈ ℤ 11 89 11 7921 Sum of Squares of First n Fibonacci Numbers = : 1 = 1 1 1 1 𝟐𝟐 1 𝒏𝒏 𝟏𝟏 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 = : 1 + 1 = 2 2 1 2 1 𝒏𝒏 𝟐𝟐 3 2 3 4 = : 1 + 1 + 4 = 6 4 3 4 9 𝒏𝒏 𝟑𝟑 5 5 5 25 = : 1 + 1 + 4 + 9 = 15 6 8 6 64 𝒏𝒏 𝟒𝟒 = : 1 + 1 + 4 + 9 + 25 = 40 7 13 7 169 8 21 8 441 𝒏𝒏 𝟓𝟓 9 34 9 1156 General Formula: 𝒏𝒏 = ⋮ 10 55 10 3025 𝟐𝟐 � 𝑭𝑭𝒊𝒊 𝑭𝑭𝒏𝒏𝑭𝑭𝒏𝒏+𝟏𝟏 11 89 11 7921 𝒊𝒊=𝟏𝟏 Why? Check Out Fibonacci Squares! Proof (Mathematical Induction)

General Formula: 𝑛𝑛 = 𝑘𝑘+1 = 𝑘𝑘 + 2 2 2 2 � 𝐹𝐹𝑖𝑖 𝐹𝐹𝑛𝑛𝐹𝐹𝑛𝑛+1 � 𝐹𝐹𝑖𝑖 � 𝐹𝐹𝑖𝑖 𝐹𝐹𝑘𝑘+1 𝑖𝑖=1 𝑖𝑖=1 𝑖𝑖=1 = : = 1 = 1 = = 1 1 = 1 = + 2 2 2 1 1 2 𝑘𝑘 𝑘𝑘+1 𝑘𝑘+1 𝒏𝒏 𝟏𝟏 𝐹𝐹 𝐹𝐹 𝐹𝐹 � = 𝐹𝐹 𝐹𝐹 + 𝐹𝐹 Suppose that = 𝑘𝑘 𝑘𝑘+1 𝑘𝑘 𝑘𝑘+1 2 𝐹𝐹= 𝐹𝐹 𝐹𝐹 � 𝐹𝐹𝑖𝑖 𝐹𝐹𝑘𝑘𝐹𝐹𝑘𝑘+1 𝑖𝑖=1 𝑭𝑭𝒌𝒌+𝟏𝟏𝑭𝑭𝒌𝒌+𝟐𝟐 for some integer 1. QED  𝑘𝑘 ≥ Fibonacci Numbers and Multiples

Every 3rd exclusively is a multiple of 2 (even) 1 1 11 89 21 10946 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 2 1 12 144 22 17711 3 2 13 233 23 28657 4 3 14 377 24 46368 5 5 15 610 25 75025 6 8 16 987 26 121393 7 13 17 1597 27 196418 8 21 18 2584 28 317811 9 34 19 4181 29 514229 10 55 20 6765 30 832040 Fibonacci Numbers and Multiples

Every 3rd Fibonacci Number exclusively is a multiple of 2 (even) 1 1 11 89 21 10946 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 2 1 12 144 22 17711 Every 4th Fibonacci Number exclusively is a multiple of 3 3 2 13 233 23 28657 4 3 14 377 24 46368 5 5 15 610 25 75025 6 8 16 987 26 121393 7 13 17 1597 27 196418 8 21 18 2584 28 317811 9 34 19 4181 29 514229 10 55 20 6765 30 832040 Fibonacci Numbers and Multiples

Every 3rd Fibonacci Number exclusively is a multiple of 2 (even) 1 1 11 89 21 10946 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 2 1 12 144 22 17711 Every 4th Fibonacci Number exclusively is a multiple of 3 3 2 13 233 23 28657 4 3 14 377 24 46368 5 5 15 610 25 75025 Every 5th Fibonacci Number exclusively is a multiple of 5 6 8 16 987 26 121393 7 13 17 1597 27 196418 8 21 18 2584 28 317811 9 34 19 4181 29 514229 10 55 20 6765 30 832040 Fibonacci Numbers and Multiples

Every 3rd Fibonacci Number exclusively is a multiple of 2 (even) 1 1 11 89 21 10946 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 2 1 12 144 22 17711 Every 4th Fibonacci Number exclusively is a multiple of 3 3 2 13 233 23 28657 4 3 14 377 24 46368 5 5 15 610 25 75025 Every 5th Fibonacci Number exclusively is a multiple of 5 6 8 16 987 26 121393 7 13 17 1597 27 196418 8 21 18 2584 28 317811 Every 6th Fibonacci Number exclusively is a multiple of 8 9 34 19 4181 29 514229 10 55 20 6765 30 832040 Fibonacci Numbers and Multiples

Every 3rd Fibonacci Number exclusively is a multiple of 2 (even) 1 1 11 89 21 10946 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 2 1 12 144 22 17711 Every 4th Fibonacci Number exclusively is a multiple of 3 3 2 13 233 23 28657 4 3 14 377 24 46368 5 5 15 610 25 75025 Every 5th Fibonacci Number exclusively is a multiple of 5 6 8 16 987 26 121393 7 13 17 1597 27 196418 8 21 18 2584 28 317811 Every 6th Fibonacci Number exclusively is a multiple of 8 9 34 19 4181 29 514229 10 55 20 6765 30 832040 th Every Fibonacci Number exclusively is a multiple of 𝒏𝒏 𝑭𝑭𝒏𝒏 Fibonacci Numbers and Divisibility

377 = = 29 13 𝐹𝐹14 1 1 11 89 21 10946 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 7 𝐹𝐹 10946 2 1 12 144 22 17711 = = 842 13 3 2 13 233 23 28657 𝐹𝐹21 7 4 3 14 377 24 46368 𝐹𝐹 317811 = = 24447 5 5 15 610 25 75025 13 𝐹𝐹28 6 8 16 987 26 121393 7 𝐹𝐹 317811 7 13 17 1597 27 196418 = = 843 377 8 21 18 2584 28 317811 𝐹𝐹28 9 34 19 4181 29 514229 𝐹𝐹14 | iff | 10 55 20 6765 30 832040 𝑭𝑭𝒎𝒎 𝑭𝑭𝒏𝒏 𝒎𝒎 𝒏𝒏 (read divides if and only if divides )

𝑭𝑭𝒎𝒎 𝑭𝑭𝒏𝒏 𝒎𝒎 𝒏𝒏 Fibonacci Numbers and Divisibility

• In case you’re curious…

• If | , then:

𝑎𝑎 𝑛𝑛

= + + + + 𝑛𝑛 𝑛𝑛 𝐹𝐹 2 ⁄𝑎𝑎 𝐹𝐹𝑛𝑛+1−𝑎𝑎 𝐹𝐹𝑎𝑎−1𝐹𝐹𝑛𝑛+1−2𝑎𝑎 𝐹𝐹𝑎𝑎−1 𝐹𝐹𝑛𝑛+1−3𝑎𝑎 ⋯ 𝐹𝐹𝑎𝑎−1 𝐹𝐹1 𝐹𝐹𝑎𝑎 Fibonacci Numbers and Primes

If is prime, then is prime, except = 3

𝑛𝑛 4 𝐹𝐹 𝑛𝑛 𝐹𝐹 1 1 11 89 21 10946 Converse NOT true! = 4181 = 113 37 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 2 1 12 144 22 17711 𝐹𝐹19 � Contrapositive: If is composite, then is 3 2 13 233 23 28657 composite, except = 3 4 3 14 377 24 46368 𝑛𝑛 𝐹𝐹𝑛𝑛 𝐹𝐹4 5 5 15 610 25 75025 We know from Discrete Mathematics that 6 8 16 987 26 121393 there are infinitely many prime numbers, but it is still unknown whether there are infinitely 7 13 17 1597 27 196418 many prime Fibonacci Numbers 8 21 18 2584 28 317811 9 34 19 4181 29 514229

As of March 2017, there are only 34 known 10 55 20 6765 30 832040 prime Fibonacci Numbers, the largest of which is (21,925 digits!)

𝐹𝐹104911 Fibonacci Numbers and Primes

Every prime divides a Fibonacci Number 1 1 11 89 21 10946 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 If a ±1 mod 10 2 1 12 144 22 17711 (i.e. ends in 1 or 9), then | 3 2 13 233 23 28657 𝑝𝑝 ≡ 𝑝𝑝 𝐹𝐹𝑝𝑝−1 4 3 14 377 24 46368 If a prime number ±3 mod 10 5 5 15 610 25 75025 (i. e. ends in 3 or 7), then | 𝑝𝑝 ≡ 6 8 16 987 26 121393 𝑝𝑝 𝐹𝐹𝑝𝑝+1 7 13 17 1597 27 196418 If 4, + 1 is always composite 8 21 18 2584 28 317811 𝑛𝑛 ≥ 𝐹𝐹𝑛𝑛 9 34 19 4181 29 514229 If 7, 1 is always composite 10 55 20 6765 30 832040 𝑛𝑛 ≥ 𝐹𝐹𝑛𝑛 − Fibonacci Numbers and GCDs gcd 8, 12 = 4 1 1 11 89 21 10946 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 gcd , = gcd 21, 144 = 3 = 2 1 12 144 22 17711 𝐹𝐹8 𝐹𝐹12 𝐹𝐹4 3 2 13 233 23 28657 gcd , = gcd ( , ) 4 3 14 377 24 46368 𝒎𝒎 𝒏𝒏 𝑭𝑭𝒎𝒎 𝑭𝑭𝒏𝒏 𝑭𝑭 5 5 15 610 25 75025 gcd , = 6 8 16 987 26 121393 𝑭𝑭𝒏𝒏 𝑭𝑭𝒏𝒏+𝟏𝟏 𝟏𝟏 7 13 17 1597 27 196418 gcd , = 8 21 18 2584 28 317811 𝑭𝑭𝒏𝒏 𝑭𝑭𝒏𝒏+𝟐𝟐 𝟏𝟏 9 34 19 4181 29 514229 , and , are 10 55 20 6765 30 832040 relatively prime 𝑭𝑭𝒏𝒏 𝑭𝑭𝒏𝒏+𝟏𝟏 𝑭𝑭𝒏𝒏 𝑭𝑭𝒏𝒏+𝟐𝟐 Fibonacci Numbers and 0.011235955

Arrange the Fibonacci Numbers in such a way 1 st that the units digit of Fn is in the (n + 1) decimal place, then add them up: 89 ≈ 0.01 0.001 0.0002 0.00003 0.000005 0.0000008 0.00000013 0.000000021

⋮ Fibonacci Numbers and 0.011235955

100 1 1.1235995 through 89 ≈ 𝐹𝐹1 𝐹𝐹5 89 ≈ Add two 0’s to the numerator and a 9 to the front and end of the denominator:

10000 1.0102030508132134559 9899 ≈ through in pairs

𝐹𝐹1 𝐹𝐹10 American mathematicians James Smoak and Thomas J. Osler proved that the pattern continues! Fibonacci Numbers and Irrationals

1 ! = ∞ 𝐹𝐹𝑖𝑖 1 +∑𝑖𝑖=1 𝑖𝑖 −+ 1 ! 𝑒𝑒 ∞ 𝐹𝐹𝑖𝑖 ∑𝑖𝑖=1 𝑖𝑖 In 1985, Yuri V. Matiyasevich of St. Petersburg, Russia developed a formula for in terms of Fibonacci Numbers: π 6 log … = lim log lcm , , … , 𝐹𝐹1𝐹𝐹2 𝐹𝐹𝑛𝑛 𝜋𝜋 𝑛𝑛→∞ 𝐹𝐹1 𝐹𝐹2 𝐹𝐹𝑛𝑛 Fibonacci Numbers and The It appears that lim . / / 𝑛𝑛+1 𝐹𝐹 1 𝒏𝒏1 𝒏𝒏+𝟏𝟏 𝒏𝒏 11 𝒏𝒏89 𝒏𝒏+𝟏𝟏. 𝒏𝒏 𝑛𝑛→∞ ≈ 𝟏𝟏 𝟔𝟔𝟔𝟔𝟔𝟔 𝒏𝒏 𝑭𝑭 𝑭𝑭 𝑭𝑭 𝒏𝒏 𝑭𝑭 𝑭𝑭 𝑭𝑭 𝐹𝐹𝑛𝑛 2 1 12 144 . Let = lim 𝟏𝟏 ≈ 𝟏𝟏 𝟔𝟔𝟔𝟔𝟔𝟔 3 2 . 13 233 . 𝐹𝐹𝑛𝑛+1 𝟐𝟐 𝟏𝟏 𝟔𝟔𝟔𝟔𝟔𝟔𝟔𝟔𝟓𝟓� 𝑟𝑟 𝑛𝑛→∞ 𝑛𝑛 4 3 . 14 377 . + 𝐹𝐹 1 𝟏𝟏 𝟓𝟓 ≈ 𝟏𝟏 𝟔𝟔𝟔𝟔𝟔𝟔 5 5 . 15 610 . = lim = lim 1 + = 1 + 𝟏𝟏 𝟔𝟔� ≈ 𝟏𝟏 𝟔𝟔𝟔𝟔𝟔𝟔 𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛−1 𝐹𝐹𝑛𝑛−1 6 8 . 16 987 . 𝑟𝑟 𝑛𝑛→∞ 𝑛𝑛→∞ 𝟏𝟏 𝟔𝟔 ≈ 𝟏𝟏 𝟔𝟔𝟔𝟔𝟔𝟔 𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛 𝑟𝑟 7 13 . 17 1597 . 1 ± 5 𝟏𝟏 𝟔𝟔𝟔𝟔𝟔𝟔 ≈ 𝟏𝟏 𝟔𝟔𝟔𝟔𝟔𝟔 = + , whose roots are 2 8 21 . 18 2584 . 𝟐𝟐 ≈ 𝟏𝟏 𝟔𝟔𝟔𝟔𝟔𝟔 ≈ 𝟏𝟏 𝟔𝟔𝟔𝟔𝟔𝟔 9 34 . 19 4181 . 𝒓𝒓 𝒓𝒓 𝟏𝟏 ≈ 𝟏𝟏 𝟔𝟔𝟔𝟔𝟔𝟔 ≈ 𝟏𝟏 𝟔𝟔𝟔𝟔𝟔𝟔 + 10 55 . 20 6765 . The positive root = 1.618 ≈ 𝟏𝟏 𝟔𝟔𝟔𝟔𝟔𝟔 ≈ 𝟏𝟏 𝟔𝟔𝟔𝟔𝟔𝟔 𝟏𝟏 𝟓𝟓 𝟏𝟏 𝟔𝟔𝟏𝟏𝟏𝟏 ≈ 𝟏𝟏 𝟔𝟔𝟔𝟔𝟔𝟔 is known as The Golden𝝓𝝓 Ratio ≈ 𝟐𝟐 The Golden Ratio

Infinite Nested Square Roots Infinite Continued Fractions 1 = 1 + 1 1 + = 1 + 1 + 1 + 1 + 1 𝑟𝑟 1 + 1 1 + 𝑟𝑟 ⋯ 1 +

1 ⋯ = 1 + 1 + 1 + 1 + = 1 + 2 𝑟𝑟 ⋯ 𝑟𝑟 = +𝑟𝑟 = + 𝟐𝟐 𝟐𝟐 𝒓𝒓 𝒓𝒓 𝟏𝟏 𝒓𝒓 𝟏𝟏 𝒓𝒓 Fibonacci Numbers and The Golden Ratio 1 = / / 𝟏𝟏 1 1 1 1 11 89 ~1.618 1 + = 𝒏𝒏 𝑭𝑭𝒏𝒏 𝑭𝑭𝒏𝒏+𝟏𝟏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 𝑭𝑭𝒏𝒏+𝟏𝟏 𝑭𝑭𝒏𝒏 1 2 1 2 12 144 . 𝟐𝟐 3 2 1.5 13 233 ~1.618 1 𝟏𝟏 𝟔𝟔𝟔𝟔𝟔𝟔𝟔𝟔𝟓𝟓� 1 + = 1 + 1 4 3 . 14 377 ~1.618 𝟑𝟑 5 5 1.6 15 610 ~1.618 𝟏𝟏 𝟔𝟔� 1 𝟐𝟐 1 + = 6 8 1.625 16 987 ~1.618 1 1 + 1 + 1 𝟓𝟓 7 13 ~1.615 17 1597 ~1.618 𝟑𝟑 8 21 ~1.619 18 2584 ~1.618 1 1 + = 9 34 ~1.618 19 4181 ~1.618 1 1 + 1 𝟖𝟖 10 55 . 20 6765 ~1.618 1 + 1 + 1 𝟓𝟓 𝟏𝟏 𝟔𝟔𝟏𝟏𝟏𝟏

⋮ Fibonacci Numbers and Linear

1 1 Consider the matrix F = = 1 0 𝐹𝐹2 𝐹𝐹1 𝐹𝐹1 𝐹𝐹0 2 1 F = = 1 1 2 𝐹𝐹3 𝐹𝐹2 𝐹𝐹2 𝐹𝐹1 3 2 F = = 2 1 3 𝐹𝐹4 𝐹𝐹3 𝐹𝐹3 𝐹𝐹2 5 3 F = = 3 2 4 𝐹𝐹5 𝐹𝐹4 𝐹𝐹4 𝐹𝐹3 F = ⋮ 𝒏𝒏 𝑭𝑭𝒏𝒏+𝟏𝟏 𝑭𝑭𝒏𝒏 𝑭𝑭𝒏𝒏 𝑭𝑭𝒏𝒏−𝟏𝟏 Fibonacci Numbers and Linear Algebra

1 1 What are the eigenvalues of F = ? 1 0

Find eigenvalues by setting det F I = 0 and solving for : − 𝜆𝜆 𝜆𝜆 1 1 = 0 1 − 𝜆𝜆 −𝜆𝜆 = 𝟐𝟐 𝝀𝝀 − 𝝀𝝀 − 𝟏𝟏 𝟎𝟎 1 ± 5 So = , the positive of which is 2 𝜆𝜆 The Golden Ratio! Fibonacci Numbers and Pythagorean Triples

Starting with = 5, every Fibonacci Number with an odd index (i.e. every𝐹𝐹5 other Fibonacci Number) represents the hypotenuse in a Pythagorean Triple (discovered by the mathematician Charles Raine)! Fibonacci Numbers and Pythagorean Triples

For all integers 3,

𝑛𝑛=≥ 2 2 𝑎𝑎𝑛𝑛 =𝐹𝐹2𝑛𝑛 − 𝐹𝐹𝑛𝑛−1 𝑏𝑏𝑛𝑛 = 𝐹𝐹𝑛𝑛𝐹𝐹𝑛𝑛−1 𝑐𝑐𝑛𝑛 𝐹𝐹2𝑛𝑛−1 For any four consecutive Fibonacci Numbers , , , ,

𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛+1=𝐹𝐹𝑛𝑛+2 𝐹𝐹𝑛𝑛+3 𝑎𝑎= 2 𝐹𝐹𝑛𝑛𝐹𝐹𝑛𝑛+3 𝑏𝑏= 𝐹𝐹𝑛𝑛++1𝐹𝐹𝑛𝑛+2 2 2 𝑐𝑐 𝐹𝐹𝑛𝑛+1 𝐹𝐹𝑛𝑛+2 Fibonacci Numbers and Pascal’s Triangle

• The sum of the numbers along the northeast diagonals of Pascal’s Triangle add to the Fibonacci Numbers!

• Discovered by Lucas in 1876:

𝑛𝑛−1 1 = 2 𝑛𝑛 − 𝑖𝑖 − 𝐹𝐹𝑛𝑛 � • Discovered by the𝑖𝑖= 0Belgian𝑖𝑖 mathematician Eugene Charles Catalan in 1846:

1 = 𝑛𝑛−1 5 2 2 + 1 𝑛𝑛 𝑖𝑖 𝐹𝐹𝑛𝑛 𝑛𝑛−1 � 𝑖𝑖=0 2𝑖𝑖 Other Explicit Formulas for Fn

• Binet’s Formula (discovered by the French mathematician Jacques Phillipe Marie Binet in 1843, among others):

1 1 + 5 1 5 = 𝑛𝑛 𝑛𝑛 5 2 2 − 𝐹𝐹𝑛𝑛 − • Trigonometric form (discovered by W. Hope-Jones in 1921):

1 = 2 cos cos 5 5 5 𝑛𝑛 𝑛𝑛 𝜋𝜋 𝑛𝑛 3𝜋𝜋 𝐹𝐹𝑛𝑛 − • And others! Other Cool Properties Is a Fibonacci Number? It is iff 5 ± 4 is a perfect square! 2 𝑛𝑛 𝑛𝑛 1 1 11 89 21 10946 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 Every positive integer can be uniquely 2 1 12 144 22 17711 expressed as the sum of distinct 3 2 13 233 23 28657 nonconsecutive Fibonacci Numbers 4 3 14 377 24 46368 5 5 15 610 25 75025 The only Fibonacci perfect squares are 6 8 16 987 26 121393 = = 1, and = 144 7 13 17 1597 27 196418 1 2 12 𝐹𝐹 𝐹𝐹 𝐹𝐹 8 21 18 2584 28 317811 The only Fibonacci perfect cubes are 9 34 19 4181 29 514229 = = 1 and = 8 10 55 20 6765 30 832040 𝐹𝐹1 𝐹𝐹2 𝐹𝐹6 And many more properties! Some Fibonacci Puzzles

Double any Fibonacci Number minus the next Fibonacci Number equals the second Fibonacci Number preceding the original 1 1 11 89 21 10946 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 2 1 12 144 22 17711 2 = for all integers 3 3 2 13 233 23 28657 Proof:𝐹𝐹𝑛𝑛 − 𝐹𝐹𝑛𝑛+1 𝐹𝐹𝑛𝑛−2 𝑛𝑛 ≥ 2 4 3 14 377 24 46368

𝐹𝐹𝑛𝑛 − 𝐹𝐹𝑛𝑛+1 5 5 15 610 25 75025 = 2 + + 6 8 16 987 26 121393 𝐹𝐹𝑛𝑛−1 𝐹𝐹𝑛𝑛−2 − 𝐹𝐹𝑛𝑛 𝐹𝐹𝑛𝑛−1 7 13 17 1597 27 196418 = + 8 21 18 2584 28 317811 2𝐹𝐹𝑛𝑛−2 𝐹𝐹𝑛𝑛−1 − 𝐹𝐹𝑛𝑛 = 2 + + 9 34 19 4181 29 514229

𝐹𝐹𝑛𝑛−2 𝐹𝐹𝑛𝑛−1 − 𝐹𝐹𝑛𝑛−1 𝐹𝐹𝑛𝑛−2 10 55 20 6765 30 832040 =

QED𝑭𝑭𝒏𝒏− 𝟐𝟐 Some Fibonacci Puzzles

For any four consecutive Fibonacci numbers, the positive difference of the 1 1 11 89 21 10946 squares of the middle two equals the 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 product of the first and last 2 1 12 144 22 17711 3 2 13 233 23 28657 = 4 3 14 377 24 46368 2 2 + 5 5 15 610 25 75025 Proof:𝑛𝑛+2 𝑛𝑛+1 𝑛𝑛 𝑛𝑛+3 𝐹𝐹 − 𝐹𝐹 𝐹𝐹 𝐹𝐹 ∀𝑛𝑛 ∈ ℤ 6 8 16 987 26 121393 2 2 7 13 17 1597 27 196418 𝐹𝐹𝑛𝑛+2 − 𝐹𝐹𝑛𝑛+1 = + 8 21 18 2584 28 317811 9 34 19 4181 29 514229 𝐹𝐹𝑛𝑛+2 − 𝐹𝐹𝑛𝑛+1 𝐹𝐹𝑛𝑛+2 𝐹𝐹𝑛𝑛+1 = 10 55 20 6765 30 832040

QED𝑭𝑭𝒏𝒏𝑭𝑭 𝒏𝒏+𝟑𝟑 Some Fibonacci Puzzles

• The sum of any ten consecutive Fibonacci 1 1 11 89 21 10946 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 𝒏𝒏 𝑭𝑭𝒏𝒏 numbers is 11 times the 2 1 12 144 22 17711 seventh 3 2 13 233 23 28657 • Holds true for any Gibonacci 4 3 14 377 24 46368 Sequence (generalized 5 5 15 610 25 75025 Fibonacci Sequence, a term coined by A.T. Benjamin and 6 8 16 987 26 121393 J.J. Quinn) 7 13 17 1597 27 196418 8 21 18 2584 28 317811 • Discovered by R.V. Heath in 9 34 19 4181 29 514229 1950 10 55 20 6765 30 832040 Proof The sum of any ten consecutive numbers of any Gibonacci Sequence is 11 times the seventh = = Consider ten consecutive numbers of a Gibonacci 1 1 Sequence , , , 𝑎𝑎 = 𝑎𝑎 + 𝑎𝑎2 = 𝑎𝑎2 + 2 𝑎𝑎1 𝑎𝑎2 ⋯ 𝑎𝑎10 3 1 2 Express each in terms of and , 1 10 𝑎𝑎 = 𝑎𝑎2 +𝑎𝑎3 4 1 2 𝑎𝑎𝑖𝑖 𝑎𝑎1 𝑎𝑎2 ≤ 𝑖𝑖 ≤ 𝑎𝑎 = 𝑎𝑎3 + 5𝑎𝑎 𝑎𝑎5 = 5𝑎𝑎1 + 8𝑎𝑎2 10 = 55 + 88 𝑎𝑎6 = 8𝑎𝑎1 + 13𝑎𝑎2 � 𝑎𝑎𝑖𝑖 𝑎𝑎1 𝑎𝑎2 7 1 2 𝑖𝑖=1 𝑎𝑎 = 13𝑎𝑎 + 21𝑎𝑎 = 11 5 + 8 = 𝑎𝑎8 = 21𝑎𝑎1 + 34𝑎𝑎2 9 1 2 𝑎𝑎1 𝑎𝑎2 𝟏𝟏𝟏𝟏𝒂𝒂𝟕𝟕 𝑎𝑎 𝑎𝑎 𝑎𝑎 QED  𝑎𝑎10 𝑎𝑎1 𝑎𝑎2 FIBONACCI NUMBERS IN PLACES YOU’D NEVER EXPECT! This is really remarkable! Fibonacci Numbers in Nature (Flowers)

Enchanter’s Nightshade: 2 petals Sego Lily: 3 petals Fibonacci Numbers in Nature (Flowers)

Wild Rose: 5 petals Lesser Celendine: 8 petals Fibonacci Numbers in Nature (Flowers)

Yellow Chamomile: Daisy: 34 petals 13 aqua spirals, 21 blue spirals Fibonacci Numbers in Nature (Flowers)

55 petals 89 petals Fibonacci Numbers in Nature (Artichokes)

• 3 or 5 clockwise spirals

• 5 or 8 counterclockwise spirals Fibonacci Numbers in Nature (Pineapples) • The scales of a pineapple form three different spiral patterns

• The number of spirals are the adjacent Fibonacci numbers 8, 13, and 21

• According to the 1977 Yearbook of Science and the Future, a study of 2000 pineapples confirmed this Fibonacci pattern! Fibonacci Numbers in Nature (Pinecones) • steep spirals

𝐹𝐹𝑛𝑛 • or gradual spirals

𝐹𝐹𝑛𝑛−1 𝐹𝐹𝑛𝑛−2 • An investigation of 4290 pinecones from 10 species of pine trees in California revealed that only 74 pinecones (~1.7%) deviated from this Fibonacci pattern! Fibonacci Numbers in Nature (Sunflowers) • The seeds of a sunflower are tightly packed in two distinct spirals going in opposite directions from the center of the head to the outer edge, one clockwise and the other counterclockwise

• Studies have shown that the number of spirals are typically adjacent Fibonacci numbers, with 21, 34, 55, 89, or 144 clockwise, paired respectively with 34, 55, 89, 144, or 233 counterclockwise Fibonacci Numbers in Nature (Bees)

• A male bee (drone) comes from an unfertilized egg, so it has a mother but no father • A female bee comes from a fertilized egg, so it has a mother and a father • The genealogy of a male bee follows the Fibonacci Sequence Fibonacci Numbers in Nature (Bees)

Male Generation Female Bees Male Bees Total Bees 1 0 1 1 2 1 0 1 Female 3 1 1 2 4 2 1 3 5 3 2 5 Female Male 6 5 3 8 7 8 5 13 8 13 8 21 Female Male Female 9 21 13 34 10 34 21 55

𝑛𝑛 𝐹𝐹𝑛𝑛−1 𝐹𝐹𝑛𝑛−2 𝐹𝐹𝑛𝑛 Fibonacci Numbers in Architecture

Turku Energia Power Plant Building in Sweden Turku, Finland Fibonacci Numbers in Architecture

Lindenbrauerei Mole Antonelliana , Germany Turin, Italy Fibonacci Numbers in Architecture

Zürich Hauptbahnhof Train Station Zürich Hauptbahnhof Train Station Zürich, Switzerland Zürich, Switzerland Fibonacci Numbers in Music

• Octave on a piano keyboard (8 notes):

• 8 white keys (whole steps)

• 5 black keys (half steps) in 2 groups of 2 keys and 3 keys

• 13 total keys Fibonacci Numbers in Poetry

• A limerick is a poem with 5 lines of the following structure:

• The (longer) first, second, and fifth lines rhyme with 3 beats (“feet”) each (~8 syllables each)

• The (shorter) third and fourth lines rhyme with 2 beats (“feet”) each (~5 syllables each)

• AABBA rhyme scheme

• Limericks whose first, second, and fifth lines have exactly 8 syllables each and whose third and fourth lines have exactly 5 syllables each have a total of 34 syllables Fibonacci Numbers in Poetry I think Fibonacci is fun. It starts with a 1 and a 1. Then 2, 3, 5, 8. But don’t stop there, mate. The fun has just barely begun!

Dr. Arthur Benjamin The Magic of Math: Solving for x and Figuring Out Why page 120 Fibonacci Numbers in Meteorology

“The record breaking low pressure system now traversing the central CONUS and the simulated IR imagery beautifully pairs with a Fibonacci Spiral. Just another example how math and meteorology go hand in hand”

-Facebook post by WxRisk.com on March 14, 2019 ( Day!)

π Speaking of Facebook…

Facebook post by Awesome Math Teachers on October 26, 2019 (3 days after Mole Day!) Fibonacci Numbers in Comics Fibonacci Numbers in Comics Fibonacci Numbers in Comics Fibonacci Numbers in Comics Fibonacci Numbers in the State of Illinois (Fillinoissi?) • Achieved statehood on December 3, 1818 • 5th most populous state according to the latest US Census in 2010 (behind CA, TX, NY, FL)

• 5 professional sports teams – all in Chicago (White Sox, Cubs, Bulls, Bears, Blackhawks)

• 8 letters (1 N, 1 O, 1 S, 2 L’s, 3 I’s) Fibonacci Numbers in the State of Illinois (Fillinoissi?) • Population according to the 2010 US

rounds to 13 million Census was ≈12.8 million, which • 13th state alphabetically

• 21st state admitted to the Union

• Interstate 55 begins in Chicago, IL and roughly follows the 89th parallel to New Orleans, LA

• BONUS: Area code 618 in southern Illinois and zip codes beginning in 618 in Champaign, Vermilion, Piatt, and Dewitt counties (first 3 digits following the decimal point of The Golden Ratio) Interested for More?

• Fibonacci Numbers are so fascinating (and that’s no fib) that a whole mathematics journal is entirely devoted to it!

• In 1963, mathematicians Verner Emil Hoggart and Brother Alfred Brousseau formed The Fibonacci Association “in order to exchange ideas and stimulate research in Fibonacci numbers and related topics”

• Since 1963 to this day, The Fibonacci Association produces a journal called The , which has grown into a well-recognized journal in

• As Brother Brousseau put it: “We got a group of people together in 1963, and just like a bunch of nuts, we started a mathematics magazine” Sources

1. Angel, Allen R.; Abbott, Christine D.; Runde, Dennis C. (2017). A Survey of Mathematics with Applications (10th ed.). New York, NY: Pearson. 2. Benjamin, Arthur (2015). The Magic of Math: Solving for x and Figuring Out Why. New𝐹𝐹 York,7 NY: Basic Books. 3. Devlin, Keith J. (2017). Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World. Princeton, NJ: Princeton University Press.

4. Fibonacci. Wikipedia. Wikimedia Foundation. https://en.wikipedia.org/wiki/Fibonacci

5. Fibonacci number. Wikipedia. Wikimedia Foundation. https://en.wikipedia.org/wiki/Fibonacci_number

6. Fibonacci numbers in popular culture. Wikipedia. Wikimedia Foundation. https://en.wikipedia.org/wiki/Fibonacci_numbers_in_popular_culture

7. Fibonacci prime. Wikipedia. Wikimedia Foundation. https://en.wikipedia.org/wiki/Fibonacci_prime

8. Gardner, Martin (2009). When You Were A Tadpole And I Was A Fish: And Other Speculations About This And That. New York, NY: Hill and Wang.

9. Grimaldi, Ralph P. (2012). Fibonacci and Catalan Numbers: An Introduction. Hoboken, NJ: John Wiley & Sons, Inc.

10. Koshy, Thomas (2017). Fibonacci and Lucas Numbers with Applications (2nd ed., Vol. 1). Hoboken, NJ: John Wiley & Sons, Inc.

11. Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York, NY: Broadway Books.

12. Sigler, Laurence E. (2002). Fibonacci's Liber Abaci: A Translation into Modern English of Leonardo Pisano's Book of Calculation. New York, NY: Springer.

13. Tanton, James S. (2016). The Power of Mathematical Visualization: Course Guidebook. Chantilly, VA: The Teaching Company. THANK YOU! [email protected]